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Embedding General Conservation Constraints in Discretizations of Hyperbolic Systems on Arbitrary Meshes: A Multidimensional Framework

Rémi Abgrall, Pierre-Henri Maire, Mario Ricchiuto

TL;DR

The paper proposes a unifying framework to embed general conservation constraints into discretizations of hyperbolic systems on arbitrary meshes, addressing both Eulerian and Lagrangian formulations. It shows that updating $u^{n+1}=u^n-Δt\;δu$ yields a natural graph-based notion of local conservation, with two equivalent formulations: flux-based and residual-based, the latter offering greater freedom for enforcing algebraic constraints and entropy conditions. It develops a comprehensive theory (including a Lax– Wendroff-type theorem) showing how RD schemes can be recast as finite-volume methods with explicit fluxes, and demonstrates how this structure enables robust, locally conservative discretizations on moving and nonuniform meshes. The framework is illustrated through a spectrum of classical and unconventional schemes (Roe, multi-dimensional Riemann fluxes, FE/RD hybrids, and a subface-based Lagrangian FV discretization), and is extended to mesh refinement, entropy control, and staggered/grid-motion contexts, with several open questions highlighted for ongoing research.

Abstract

The purpose of this review is to discuss the notion of conservation in hyperbolic systems and how one can formulate it at the discrete level depending on the solution representation of the solution. A general theory is difficult. We discuss several possibilities: if the solution is represented by average in volumes; if the mesh is staggerred; if the solution is solely represented by point values and an example where all the previous options are mixed. We show how each configuration can provide, or not, enough flexibility. The discussion could be adapted to any hyperbolic system endowed with an entropy, but we focus on compressible fluid mechanics, in its Eulerian and Lagrangian formulations. The unifying element is that we systematically express the update of conserved variables as $u^{n+1}=u^n- Δt\; δu$, where the functional $δu$ depends on the value of $u$ in the stencil of the scheme. Then, one can naturally define a graph connecting the states defining $δu$. The notion of local conservation can be defined from this graph. We are aware of only two possible situations: either the graph is constructed from the faces of the mesh elements (or the dual mesh), or it is defined from the mesh itself. Two notions of local conservation then emerge: either we define a numerical flux, or we define a "residual" attached to elements and the degrees of freedom within the element. We show that this two notions are in a way equivalent, but the one with residual allows much more flexibility, especially if additional algebraic constraints must be satisfied. Examples of specific additional conservation constraints are provided to illustrate this. We also show that this notion of conservation gives a very clear framework for the design of scheme in the Lagrangian framework. We end by providing a number of ongoing research questions, and highlight some open questions.

Embedding General Conservation Constraints in Discretizations of Hyperbolic Systems on Arbitrary Meshes: A Multidimensional Framework

TL;DR

The paper proposes a unifying framework to embed general conservation constraints into discretizations of hyperbolic systems on arbitrary meshes, addressing both Eulerian and Lagrangian formulations. It shows that updating yields a natural graph-based notion of local conservation, with two equivalent formulations: flux-based and residual-based, the latter offering greater freedom for enforcing algebraic constraints and entropy conditions. It develops a comprehensive theory (including a Lax– Wendroff-type theorem) showing how RD schemes can be recast as finite-volume methods with explicit fluxes, and demonstrates how this structure enables robust, locally conservative discretizations on moving and nonuniform meshes. The framework is illustrated through a spectrum of classical and unconventional schemes (Roe, multi-dimensional Riemann fluxes, FE/RD hybrids, and a subface-based Lagrangian FV discretization), and is extended to mesh refinement, entropy control, and staggered/grid-motion contexts, with several open questions highlighted for ongoing research.

Abstract

The purpose of this review is to discuss the notion of conservation in hyperbolic systems and how one can formulate it at the discrete level depending on the solution representation of the solution. A general theory is difficult. We discuss several possibilities: if the solution is represented by average in volumes; if the mesh is staggerred; if the solution is solely represented by point values and an example where all the previous options are mixed. We show how each configuration can provide, or not, enough flexibility. The discussion could be adapted to any hyperbolic system endowed with an entropy, but we focus on compressible fluid mechanics, in its Eulerian and Lagrangian formulations. The unifying element is that we systematically express the update of conserved variables as , where the functional depends on the value of in the stencil of the scheme. Then, one can naturally define a graph connecting the states defining . The notion of local conservation can be defined from this graph. We are aware of only two possible situations: either the graph is constructed from the faces of the mesh elements (or the dual mesh), or it is defined from the mesh itself. Two notions of local conservation then emerge: either we define a numerical flux, or we define a "residual" attached to elements and the degrees of freedom within the element. We show that this two notions are in a way equivalent, but the one with residual allows much more flexibility, especially if additional algebraic constraints must be satisfied. Examples of specific additional conservation constraints are provided to illustrate this. We also show that this notion of conservation gives a very clear framework for the design of scheme in the Lagrangian framework. We end by providing a number of ongoing research questions, and highlight some open questions.

Paper Structure

This paper contains 50 sections, 11 theorems, 395 equations, 15 figures.

Table of Contents

  1. Introduction
  2. First notations.
  3. Mathematical and physical setting
  4. Hyperbolic problems.
  5. The Euler equations of compressible fluid dynamics.
  6. The Lagrangian formalism
  7. The Eulerian formalism
  8. Non conservative forms of the Euler equations
  9. The Riemann problem.
  10. Several motivating examples
  11. The one dimensional case
  12. Simple solver
  13. Example of the Roe scheme
  14. Finite volume schemes I: formulation with two-states Riemann fluxes
  15. Finite volume methods II: multi-dimensional Riemann fluxes
  16. ...and 35 more sections

Key Result

Theorem 2.1

Let $\eta({\mathbf{{u}}})$ be a strictly concave entropy function for hyper; then the entropy variable ${\mathbf{{u}}}\mapsto \bm{w}=\nabla_{\mathbf{{u}}} \eta({\mathbf{{u}}})$ defines a one-to-one mapping and allows to symmetrize hyper.

Figures (15)

  • Figure 1: Notations for the finite volume schemes. On the left: definition of the control volume for the degree of freedom $\sigma$. The vertex $\sigma$ plays the role of the vertex $1$ on the left picture, etc for the triangle K. The control volume $C_\sigma$ associated to $\sigma=1$ is green on the right and corresponds to $1IGK$ on the left. The vectors ${\mathbf{{n}}}_{ij}$ are normal to the internal edges scaled by the corresponding edge length
  • Figure 2: Geometry of a polygonal grid: primal and dual cells.
  • Figure 3: Sub-cell and reversed normals.
  • Figure 4: Multidimensional upwinding.
  • Figure 5: A refaire Dotted line: primal mesh with element $\alpha$, Plain lines: dual mesh with elements $\beta$. The element $\beta'$ shares with $\beta$ the face containing $\sigma_1$ and $\sigma_2$.
  • ...and 10 more figures

Theorems & Definitions (30)

  • Theorem 2.1
  • Remark 2.1: Gamma gas law
  • Remark 2.2: Gibbs relation
  • Remark 2.3: Calorically perfect gas
  • Remark 2.4: Geometrical Conservation Law
  • Definition 3.1
  • Remark 3.1
  • Remark 3.2: Petrov Galerkin formulation of the limited RD scheme
  • Remark 3.3: About the invariance domain property of the N scheme
  • Lemma 4.1
  • ...and 20 more